3.41 \(\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx\)

Optimal. Leaf size=8 \[ -\frac {1}{2} \cot ^{-1}(x)^2 \]

[Out]

-1/2*arccot(x)^2

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4885} \[ -\frac {1}{2} \cot ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/(1 + x^2),x]

[Out]

-ArcCot[x]^2/2

Rule 4885

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(a + b*ArcCot[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx &=-\frac {1}{2} \cot ^{-1}(x)^2\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ -\frac {1}{2} \cot ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x]/(1 + x^2),x]

[Out]

-1/2*ArcCot[x]^2

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fricas [A]  time = 0.64, size = 6, normalized size = 0.75 \[ -\frac {1}{2} \, \operatorname {arccot}\relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/(x^2+1),x, algorithm="fricas")

[Out]

-1/2*arccot(x)^2

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giac [A]  time = 0.11, size = 8, normalized size = 1.00 \[ -\frac {1}{2} \, \arctan \left (\frac {1}{x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/(x^2+1),x, algorithm="giac")

[Out]

-1/2*arctan(1/x)^2

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maple [A]  time = 0.04, size = 7, normalized size = 0.88 \[ -\frac {\mathrm {arccot}\relax (x )^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x)/(x^2+1),x)

[Out]

-1/2*arccot(x)^2

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maxima [A]  time = 0.31, size = 6, normalized size = 0.75 \[ -\frac {1}{2} \, \operatorname {arccot}\relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/(x^2+1),x, algorithm="maxima")

[Out]

-1/2*arccot(x)^2

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mupad [B]  time = 0.62, size = 6, normalized size = 0.75 \[ -\frac {{\mathrm {acot}\relax (x)}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(x)/(x^2 + 1),x)

[Out]

-acot(x)^2/2

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sympy [A]  time = 0.88, size = 7, normalized size = 0.88 \[ - \frac {\operatorname {acot}^{2}{\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x)/(x**2+1),x)

[Out]

-acot(x)**2/2

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